In computer aided design, a designer has the ability to verify and modify many aspects of a design prior to implementation. This is important during initial design stages as well as during redesign or modification.
Such a design process begins with requirements and drawings. The drawings allow for artistic elements to be incorporated into designs of objects such as those for automobiles, buildings, boats, etc. The drawings are then transformed into computer models using computer aided design tools. These models are analysed to ensure adherence to design parameters. Returning to our example of an automobile, static parameters such as interior space, head room, wheel base, engine room, and trunk space are evaluated. Dynamic parameters such as aerodynamics, heating and cooling fluid flow, temperature effects on the engine, and wind noise are also analysed.
A redesign process begins with an existing object or space. The object or space is modelled with the help of a computer aided design system. Such a model already exists when the original object or space was designed using computer aided design. The model of the existing object or space is modified to change appearance as desired and parameters are verified. In redesigning an automobile to improve aerodynamics--decrease drag--to increase fuel efficiency, a computer model of the automobile as it exists is created. The model is then analysed for aerodynamics. The analysis allows the designers to modify the model of the automobile in order to achieve desired results.
In analysing fluid mechanics, heat and mass transfer, physics, chemistry, solid mechanics, and structural analysis, a "grid" is generated representing a space through which fluid flows or the flux, e.g. of heat, occurs, in order to allow the problem to be discretised for the purposes of numerical calculations on a digital computer. Such grids are known and have been used extensively in design and analysis being also referred to as "cells", "meshes" "elements" and so forth. Unfortunately, generating such grids is time consuming and each time a space or an object such as an aircraft is modified, the grid must be regenerated. Typically, designs requiring analysis using computational fluid dynamics require a 3 stage cycle comprising: grid generation (pre-processing), flow solving, and post-processing. The cycle is repeated a number of times in obtaining a refined analysis of the flow. Further, each stage requires a distinct suite of software.
The dominant type of grid in use today is the structured body-fitted coordinate (BFC) grid composed of quadrilateral (two-dimensional grid) or hexahedral elements (three-dimensional grid) cells. The grid-generation process involves the definition of two or three functions, .xi..sup.i, (also denoted by .xi., .eta., .zeta.). These are considered a function of the Cartesian components, x.sup.i, also denoted by x, y, z, i.e. .xi..sup.i =.xi..sup.i (x.sup.1, x.sup.2, x.sup.3) i=1, 2, 3. Relationships between (.xi..sup.1, .xi..sup.2, .xi..sup.3) and (x.sup.1, x.sup.2, x.sup.3) are often stipulated by means of partial differential equations, ##EQU1## where the symbol D denotes a differential operator. It is, however, the Cartesian coordinates of the grid-points, x.sup.i =x.sup.i (.xi..sup.1, .xi..sup.2, .xi..sup.3) which are required. Hence, an inverse form ##EQU2## is solved.
A conventional grid generation process involves the use of transfinite interpolation to obtain initial grid values x.sup.i. This is then followed by `relaxation` or `smoothing` based on the solution of equation 2.
The D-operator is expressible in terms of vector functions so, choice of independent variables is inconsequential and equation 1 may be replaced by a coordinate independent form, EQU D(.phi.)=0, .phi.=.xi..sup.i ( 3)
Among the simplest and most widely-used grids are those based on Laplace's equation, which in the operator form of equation 3 is, EQU D(.phi.)=.gradient..multidot..gradient.(.phi.)=0 (4)
where .phi.=.xi..sup.1, .xi..sup.2, .xi..sup.3. A coordinate dependent form of equation 4 is, ##EQU3## which is inverted as, ##EQU4## g.sup.jk are the contravariant components of a metric tensor expressed in tensor notation.
Another common set of equations used in grid generation are Poisson's equations, which in the operator form of equation 3 are, EQU D(.phi.)=.gradient..multidot..gradient..phi.-S=0 (7)
where S=S.sup.i are `control-functions` also referred to as S=-(P, Q, R). Typically, the inverse of equation 7 is solved. Various physical analogies, e.g. heat conduction with internal sources, can be constructed to provide a phenomenological basis for equation 7. "Control-functions" of the general form S=-.alpha.sign(.xi.-.xi..sub.0)e.sup.-h.vertline..xi.-.xi..vertline. may be used to effect surface attractions and have been described in the art. There are, however, disadvantages to the use of exponential control-functions: the presence of two coefficients, a and b, demands a measure of skill on the part of the programmer in order to concentrate grid cells effectively; and convergence is not guaranteed; the solution is potentially unstable. Other "automatic" control-functions based on presumed relations of boundary-orthogonality attempting to control cell distributions away from the boundaries are described in the art. These are not entirely successful in achieving the desired goal.
It would be advantageous to provide a method of grid generation that requires a simple algorithm allowing significant control of grid line locations and improved likelihood of convergence over known methods of grid generation.